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07.03, 07.05 HC)

Use the function f(x) = −16x2 + 22x + 3 to answer the questions.

Part A: Completely factor f(x). (2 points)

Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)

Part C: Describe the end behavior of the graph of f(x). Explain. (2 points)

Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to dra

Respuesta :

Lanuel
  1. The x-intercepts of the graph of f(x) are (-1/8, 3/2).
  2. The parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.
  3. Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
  4. Determine the y-intercept (3).
  5. Connect all the known points on the graph to form a downward parabola.

What is a quadratic function?

A quadratic function can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the graph of any quadratic function is parabolic because it is a u-shaped curve. For the given quadratic function, the graph is a downward parabola because the coefficient of x² is negative.

How to determine the x-intercepts?

In order to determine the x-intercepts of the graph of f(x), we would make to be y = 0.

By rearranging and completely factoring the given quadratic function, we have:

f(x) = -16x² + 22x + 3

f(x) = 16x² - 22x - 3

16x² - 24x + 2x - 3 = 0

8x(2x - 3) + 1(2x - 3) = 0

(8x + 1)(2x - 3) = 0

x₁ ⇒ 8x = -1      ⇒     x₁ = -1/8.

x₂ ⇒ 2x = 3      ⇒     x₂ = 3/2.    

Therefore, the x-intercepts of the graph of f(x) are (-1/8, 3/2).

How to describe the end behavior?

Since the parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.

V(x) = -b/2a

V(x) = -22/2(-16)

V(x) = -22/-32

V(x) = 11/16

V(y) = -D/4a

V(y) = -16(11/16)² + 22(11/16) + 3

V(y) = -16(121/256) + 121/8 + 3

V(y) = -121/16 + 121/8 + 3

V(y) = (-121 + 242 + 48)/16

V(y) = 169/16.

What are the steps you would use to graph f(x)?

  • Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
  • Determine the y-intercept (3).
  • Connect all the known points on the graph to form a downward parabola.

Read more on quadratic functions here: https://brainly.com/question/24020644

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